Summary
LetX(t) be a fractional Wiener process, i.e., a centered Gaussian process on [0, ∞) with stationary increments and varianceEX 2 (t)=t 2α, anda(t) a positive nondecreasing function witha(t)≦t. We investigate the a.s. asymptotic behaviour of the incrementsI(t, a (t))=max{X{u+a(t))−X(u): 0≦u≦t−a(t)} (and some others that are similarly defined) ast→∞.
References
Grill, K.: On the increments of the Wiener process. Submitted for publication.
Ortega, J.: Upper classes for the increments of fractional Wiener processes. Probab. Th. Rel. Fields80, 365–380 (1989)
Ortega, J., Wschebor, M.: On the increments of the Wiener process. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 329–339 (1984)
Qualls, C., Watanabe, H.: Asymptotic properties of Gaussian processes. Ann. Math. Stat.43, 580–596 (1972)
Qualls, C., Watanabe, H.: Asymptotic properties of Gaussian random fields. Trans. Am. Math. Soc.177, 155–171 (1973)
Révész, P.: A note to the Chung-Erdös-Sirao theorem. In: Chakravarty, L.M. (ed.) Asymptotic theory of statistical tests and estimation, pp 147–158. New York: Academic Press 1980
Révész, P.: On the increments of Wiener and related processes. Ann. Probab.10, 613–622 (1982)
Spitzer, F.: Principles of random walk. Princeton: Van Nostrand 1964
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grill, K. Upper classes for the increments of the fractional Wiener process. Probab. Th. Rel. Fields 87, 411–416 (1991). https://doi.org/10.1007/BF01304273
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01304273
Keywords
- Stochastic Process
- Asymptotic Behaviour
- Probability Theory
- Mathematical Biology
- Gaussian Process